Loop Quantum Gravity a la Aharonov-Bohm

Eugenio Bianchi

Centre de Physique Theorique de Luminy, Marseille

International LQG Seminar – October 6th, 2009

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 1 / 21

In LQG, the state space KDiff admits an orthogonal decompositioninto subspaces KDiff

{Γ} ( {Γ} = diff. equiv. class of spin-network graphs Γ ⊂ Σ )

Remarks: [Rovelli-Smolin, Ashtekar-Lewandowski, ∼ ’90]

(i) The state space is obtained via- loop quantization of the Ashtekar-Barbero connection- group averaging over Diff(Σ)

(ii) States in KDiff{Γ} admit a physical interpretation in terms of

quantum geometry

Aim of the talk:

show that the state space KDiff{Γ}, together with its interpreta-

tion (ii), can be obtained via standard QFT quantization ofa theory of locally-flat Ashtekar-Barbero connections withmagnetic defects

[ based on arXiv:0907.4388 ]

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 2 / 21

Plan of the talk

Motivation and the idea

The framework:- defects, locally-flat connections and diff-invariant states

- the QFT scalar product

An example: the state space for a single line defect

Relation with canonical LQG and with Spin Foams

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 3 / 21

MotivationThe mathematical structure of the Hilbert space KDiff

{Γ} and its physical inter-pretation in terms of quantum geometry are the ones proper to a systemwith a finite (but possibly large) number of degrees of freedom

Therefore, while the full theory has an infinite number of degrees of freedom(as classical GR does), each subspace captures only a finite number of them

Question:

• is there a classical stystem with a finite number of degrees of freedomsuch that, when quantized, leads directly to the state space KDiff

{Γ}?

• which degrees of freedom of General Relativity are we capturing?

Notice that Spin Foam quantization offers an answer to these questions

- discretization of GR, then quantization of the discrete system- boundary state space: spin networks with abstract graph dual to

the discretization

However, due to the discretization and to the fact that the graph is notembedded, the role of diffeomorphisms is rather obscure

Here we look for a system defined on a manifold, so that diffeomor-phisms act on local fields in the standard way, via pullback

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 4 / 21

The idea: Aharonov-Bohm setting

Consider a configuration of the connection Asuch that its magnetic field B vanishes in theregion Σ − l.

Despite the vanishing of B, there are observ-able effects associated to the holonomy ofthe connection along non-contractible closedpaths.

Such effects depend only on the flux Φ of themagnetic field enclosed by the path.

In particular, the effect does not change if thepath is deformed continuously.

l

Σ

γ

Φ

B = 0

Idea: KDiff{γ}= space of Diff-invariant states Ψγ,η[A] = η(hγ [A]) = f(Φ)

Scalar product via QFT path integral measure on the local field A(x)

Similarly, the Aharonov-Bohm effect for electrons depends only on Φand not on the specific path γ

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 5 / 21

From Feynman Lectures :

The original paper :Y. Aharonov and D. Bohm, “Significance of electromagneticpotentials in the quantum theory”, Phys. Rev. 115 (1959) 485.

e

i

~e

∫

γ

Adx= e

i 2πΦ

Φ0

θ ≈ x0/L ≈ 2π Φ

Φ0

Φ = flux of B

Φ0 = 2π~

e≈ 4×10−7G · cm2

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 6 / 21

The framework : manifold Σ′ and locally-flat connections1. Start with a 3-manifold Σ connected and without boundary

Consider a network D of curves embedded in Σ

Regard D as a network of defects and introduce a new manifold Σ′

Σ′ = Σ −D

Σ′ is path connected, but not simply connectedExample: - consider a cellular decomposition C(Σ) of the original manifold

- regard its 1-skeleton as a defect-network, D = C1(Σ)

closed paths around edges of the 1-skeleton are non-contractible⇒ non-trivial first homotopy group π1(Σ

′)

2. Introduce a locally-flat connection A(x) on Σ′, Af = {A : Σ′ → G | F (A) = 0}

Despite being locally-flat, the connection can have non-trivialholonomy around non-contractible loops in Σ′

AB effect

3. The space Af/G of locally-flat connections modulo gauge transformationsis finite dimensional. We call N this moduli space

N = Af/G = Hom(π1(Σ′), G)/G

and {mr} coordinates (moduli) on N

4. Locally-flat connection Amr,g: labeled by its moduli mr and by a gauge transf g(x)

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 7 / 21

The framework : the state space K′ and functions of moduliWe consider the kinematics of General Relativity in Ashtekar-Barbero variablesThe configuration variable is a SU(2) connection A(x) on a 3-manifold Σ

In Loop Quantum Gravity: kinematical Hilbert space K of functionals of theconnection invariant under - SU(2)-gauge transformations

- Diff(Σ) of the manifold Σ

Here we follow a similar procedure but we require also- topological invariance, F Ψ[A] = 0

We introduce a state space K′ of gauge invariant functionalsof a locally-flat connection on Σ′

∗ Notice that: in a topological gauge theory, gauge- and diff-invariance are related⇒ once we have gauge-invariance, Diff-invariance comes for free

As states in K′ are gauge-invariant, they can depend on the connection Amr,g

only through its moduli mr

Ψf [Amr,g] = f(m1, . . , mR) [a]

⇒ states labeled by a function f on moduli spacef : N → C

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 8 / 21

The framework : states in terms of dual graph and holonomiesCan we introduce an embedded graph Γ ⊂ Σ′ and write states as cylindricalfunctions? ΨΓ,η[A] = η(hγ1

[A], . . , hγL[A]) [ b ]

1. Notice that contractible loops of the graph evaluate to the identity

2. Choose minimal graph Γ′: embedded in Σ′ and “dual” to the defect-network D

3. Property the dual graph satisfies: edge-path-group π(Γ′) isomorphic to π1(Σ′)

4. Construction in a specific case: π(Γ′) = π1(Σ −D)

- Focus on a defect-network arising from a cell decomposition, D = C1(Σ)

- Consider an embedded graph Γ′ dual to the decomposition, Γ′ =(

C(Σ)∗)

1

5. Properties:- Gauge-invariance, ΨΓ′,η[Amr,g] = ΨΓ′,η[Amr,h·g]

- Diff-invariance from local-flatness: φ ∈ Diff(Σ′) connected to the identity,

ΨΓ′,η[φ∗A] = Ψφ−1◦Γ′,η[A]!= ΨΓ′,η[A]

- ΨΓ′,η[A] depends on Γ′ only via its ‘knotting’ with the defect-network D

6. Spin-network states with graph Γ′ as cyl functions labelled by Γ′ and ηjl,in

ηjl,in(h1, . . , hL) =(

⊗

n⊂Γ′ vin

)

·(

⊗

γl⊂Γ′ D(jl)(hl))

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 9 / 21

The framework : summarizingThere are two equivalent characterizations of diff-invariant states in K′:

[a] in terms of a function f on moduli space N = Hom(

π1(Σ′), SU(2)

)

/SU(2)

Ψf [Amr,g] = f(m1, . . , mR) N = {m1, . . , mR}

[b] in terms of a dual graph Γ′ and a cylindrical function η : SU(2)L → C

ΨΓ,η[A] = η(hγ1[A], . . , hγL

[A])

The characterization of states [ b ] in terms of (Γ′, η) is slightly redundant, butleads to a picture definitely more clear than the one in terms of moduli [a]

We can go from description [b] to description [a] gauge-fixing the SU(2)invariance at the nodes of Γ′

- once chosen coordinates {m1, . . , mR} on N , the couple (Γ′, η)determines a unique function f : N → C

- e.g.: choose a maximal tree of Γ′ and fix to the identitythe group elements associated to it

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 10 / 21

The framework : the scalar productIn order to promote the linear space described before to a Hilbert spacewe need to introduce a scalar productA topologically invariant functional measure D[A] has to reduce to an ordinarymeasure on moduli space N

〈f |g〉 =

∫

Af /G

D[A] Ψf [A] Ψg[A] =

∫

N

dµ(mr) f(m1, . . , mR) g(m1, . . , mR)

Making sense of the formal functional measure D[A] amounts to a choice ofmeasure dµ(mr) on the moduli space N

Problem: Is there a principle that can guide us in the choice of the measure dµ(mr)?

Here: we use the methods proper to quantum field theory to make sense of thefunctional integral over locally-flat connections modulo gauge transformations.Such methods, in the case of a finite dimensional moduli space, allow to fullydetermine a specific measure on the moduli space and answer our question

used in string measure: Polyakov 1981, Alvarez 19832+1 gravity with and w/o particles: Carlip 1995, Cantini Menotti 2002simplicial measure: Jevicki Ninomiya 1986, Menotti Peirano 1995

∗ The guiding principle here is locality, that is:

the configurations we are integrating over are local fields

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 11 / 21

The framework : the scalar product via QFT methods1. In the integral over locally-flat connections D[Amr,g(x)], we can change

variables to the moduli mr and gauge transformations g(x)

The integral over gauge transformations factorizes so that we are left with a finitedimensional integral over moduli, with a non-trivial measure on the moduli

2. Equivalently, we can use the more well-known Faddeev-Popov procedure:Amr,g = g−1Amr g + g−1dg with Amr gauge-fixed to χ(Amr ) = 0

measure on locally-flat connections modulo gauge transformations written as∫

Af /G

D[A] =

∫

A

[

∏

x

dA(x)]

δ(

F (A))

δ(

χ(A))

∆FP(A)

where χ(A) = 0 is a gauge-fixing condition and ∆FP(A) =∣

∣

∣Det

δχ

δξ

∣

∣

∣

3. Integrating and changing variables, we end up with a finite dimensional integral∫

Af /G

D[A] =

∫

dAmr ∆FP(Amr ) =

∫

N

dm1 · · · dmR J(mr)∆FP(mr)

where J(mr) is the Jacobian from gauge-fixed connections Amr to the moduli mr

4. This is our field theoretical proposal for the measure dµQFT(mr): states in theHilbert space K′ are assumed to be normalizable in the scalar product

〈f |g〉 =

∫

Af /G

D[A] Ψf [A] Ψg[A] =

∫

N

dm1 · · · dmR J(mr)∆FP(mr) f(mr) g(mr)

→ explicit expression for the measure dµQFT(mr) in our example

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 12 / 21

The framework : the scalar product via Loop methodsThe field theoretical derivation of the measure dµ(mr) described above can becompared to a different construction which is proper to Loop Quantum Gravity

1. Using the characterization of states [ b ] in terms of the dual graph Γ′, there is anatural choice of measure: the Haar measure on links of the graph

〈η|ξ〉 =

∫

Af /G

D[A] ΨΓ′,η[A] ΨΓ′,ξ[A] =

∫

SU(2)L

L∏

l=1

dµH(hl) η(h1, . . , hL) ξ(h1, . . , hL)

2. With this choice of measure, spin-network states ηjl,in provide an o.n. basis of K′

〈ηjl,in |ηj′l,i′n

〉 =(

∏

l

δjl,j′

l

)(

∏

n

δin,i′n

)

3. Notice that, as the functions η and ξ are invariant under conjugation at nodes,

the Haar measure∏

l dµH(hl) on SU(2)L reduces to a specific

measure dµ(mr) on moduli via the Weyl integration formula∗

Therefore it provides again a definition of the functional measure D[A]via a prescription of a specific measure dµLoop(mr) on moduli space.

∗[equivalent to gauge-fixing of SU(2) invari-ance at nodes via a choice of maximal tree]

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 13 / 21

The framework : the scalar product – a conjecture

- While the group theoretical choice of measure a la Loop is certainly natural,the field theoretical choice is a well-motivated one

What is the relationship between the two?

- We investigate these two proposals in a rather simple caseand provide an explicit expression for the measure dµ(mr)

- The result supports the following conjecture:

the QFT measure on the moduli space N obtained via the fieldtheoretical construction coincides with the one obtained from theproduct of Haar measures via Weyl integration formula

dµQFT(mr) = dµLoop(mr)

- If such conjecture turns out to be robust, then spin network states withembedded graph Γ′ are not just a good tool for describing Diff-invariantfunctionals of the connection but provide an o.n. basis of the Hilbertspace K′ built via the standard field theoretical construction

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 14 / 21

An example: Σ′ with a single line defect1. Original manifold with trivial topology Σ ≈ R

3

Defect-network consisting of a single line l ≈ R (assumed to be unknotted in Σ)−→ the new manifold Σ′ = Σ − l has non-trivial π1(Σ

′) generated bythe homotopy class [γ] of loops which encircle once the line l

2. SU(2) locally-flat connection A on Σ′

3. In order to identify a representative in each gauge orbit, we introduce agauge fixing condition −→ break the local symmetries of the problem:−→ a convenient choice is to introduce an auxiliary metric qab(x) on Σ

and consider the Coulomb-like gauge χ = qab∂aAb

we assume qab = δab, choose Cartesian coord xa = (x, y, z), l along the z-axes

4. Gauge-fixed locally-flat connection in Σ′

Aia(x) =

Φi

2παa(x) with αa(x) =

(

−y

x2 + y2,

x

x2 + y2, 0

)

5. its holonomy along a loop γ is simply given by

hγ [A] = exp[i(

∫

γ

αadxa)Φi

2πτi ] where

∫

γ

αadxa = 2πn

n = winding number of the loop γ around the line l

−→ the holonomy hγ [A] provides a homomorphism from π1(Σ′) to SU(2)

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 15 / 21

An example: physical interpretation of Φi and the moduli space1. As the connection is locally-flat in Σ′, the

non-abelian magnetic field Bai = 1

2εabcF i

bc

vanishes everywhere except along the line l

Bai (x) =

∫

l

ds Φi xa(s) δ(3)(x − x(s))

The line l can be thought as a thin solenoid

{Flux of Bai through S in fig} = Φi

Via Stokes theorem, it determines hγ [A]

l

Σ

S

γ

2. The magnetic flux Φi provides a parametrization of the moduli space oflocally-flat connections. To find its range we have to identify different mag-netic fields which correspond to the same holonomy ⇒ Φi ∈ S3

3. States = functions of the flux Φi. Gauge invariance ⇒ depend only on φ = |Φi|

As a result, states in K′ are labeled by a function f(φ),

Ψf [Aφ,g] = f(φ)

and the moduli space is given byN = {Φi ∈ S3}/SU(2) = {φ ∈ [0, 2π]}

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 16 / 21

An example: the QFT scalar product1. QFT measure on moduli space dµQFT(φ) = J(φ) ∆FP(φ) dφ

2. Jacobian easily determined: d3Φi = φ2dφd2vi, so that J(φ) = φ2

3. Faddeev-Popov term: determinant of the operator K(Φi)

Kij(Φi) =

δχi

δξj= −δij∆ − εijk

Φk

2π

4. Eigenvalues λn = n2 + n φ2π

with n = ±1,±2, . . . and twice degenerate

5. Φi dependence extracted considering the appropriately regularized ratio

∆FP(φ) = cDetK(Φi)

DetK(0)= c

∏∞n=1

(

λn(φ))2(

λ−n(φ))2

∏∞n=1

(

λn(0))2(

λ−n(0))2 =

= c

(

∞∏

n=1

(

1 −( φ

2πn

)2)

)2

= c

(

sin φ/2

φ/2

)2

The constant c is undetermined and is fixed so that the measure of N is one

6. Therefore: dµQFT(φ) =1

πsin2 φ

2

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 17 / 21

An example: the QFT scalar product and the Loop one1. Finally we have that the QFT scalar product on K′ defined by this measure is

〈f |g〉 =

∫

Af /G

D[A]Ψf [A]Ψg[A] =

∫

N

dφ J(φ)∆FP(φ) f(φ) g(φ) =1

π

∫ 2π

0

dφ sin2 φ

2f(φ) g(φ)

2. The QFT measure can be compared to the one obtained from thegroup theoretical construction via Weyl integration formula

3. We can label states by the homotopy class [γ] of loops encircling once the line l,and by a complex-valued function η on SU(2),

Ψγ,η[A] = η(hγ [A])

Gauge invariance at the base point of γ requires that η is a class function

4. The Loop scalar product is given by the Haar measure on SU(2) and reduces toan integral over the class angle φ/2. Defining f(φ) = η(exp iφτ3), we have

〈η|ξ〉 =

∫

Af /G

D[A] Ψγ,η[A]Ψγ,ξ[A] =

∫

SU(2)

dµ(hγ) η(hγ) ξ(hγ) =1

π

∫ 2π

0

dφ sin2 φ

2f(φ) g(φ)

which coincides with the field theoretical one derived above

5. This fact supports our conjecture and leads to a physical interpretation of theclass angle as the modulus of the flux of the magnetic field through the defect line

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 18 / 21

Relation with canonical LQG1. In ordinary LQG, embedded spin networks are not diffeomorphism invariant.

Huge non-separable Hilbert space. Most states are identified when diffeomor-phism invariance is imposed via group-averaging

2. Abstract spin networks (s-knots) depend only on the diffeomorphism equiv-alence class of the graph and on the SU(2) labels

3. The diffeomorphism equivalence class of the graph knows about:

- connectivity of the graph- its knotting with the non-trivial topology of the manifold- self-knotting of the graph- continuous parameters (Grot-Rovelli moduli) related to tangents to the

links at each node (unless Diff∗ are considered [Fairbairn-Rovelli ’04])4. On the other hand, here we focus directly on diff-invariant states

5. Thanks to local-flatness, spin networks have embedded graph and never-theless the state they define is diffeomorphism-invariant

6. No dependence on self-knotting, nor on moduli at nodes

7. States depend only on the knotting of the graph with the manifold Σ′, i.e.with the network of defects and with the topology of the original manifold Σ

8. From this point of view, they are closer to the use that is made ofspin networks as boundary states within the context of spin foams

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 19 / 21

Relation with Spin Foams1. The construction discussed is purely kinematical. A natural tool for implement-

ing its dynamics is provided by Spin Foams2. Network of defects as the 1-skeleton of a cellular decomposition of Σ. Spin

networks with graph dual to the decomposition3. Dual picture of quantum geometry: operators build out of the flux of the electric

field (through surfaces ending at defects) attach a geometric meaning to thecells of the decomposition (quanta of volume and of area) [in progress]

4. No non-local links [ cfr. Markopoulou-Smolin ’07]

5. Same structure of the Spin Foam boundary space [Lewandowski generaliz.’09]

6. Spin foam quantization [EPRL-FK, ’08]: at the classical level, GR as a topologicalBF theory with constraints on the B field; at the quantum level, constraints areimposed only on the 2-boundary of the 4-cells of a simplicial complex- building-blocks have flat connection- 2-skeleton of defects (where the constraints are imposed) support curvature

7. A foliation reproduces a theory of locally-flat connections with magnetic defects

8. In ordinary LQG: Ashtekar-Lewandowski configuration space consists ofdistributional polymeric connections; then Diffs are taken into account.Here the degrees of freedom are associated to quantum configurationsconsisting of connections which are locally-flat but have distributionalpolymeric magnetic field

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 20 / 21

Conclusions

We have considered a theory of locally-flat Ashtekar-Barbero connectionswith magnetic defects and quantized it via standard field theoretical methods

The resulting state space K′ is surprisingly close to the one of ordinary LQG

Spin networks with embedded graph Γ′ provide the appropriate tool for de-scribing Gauge- and Diff-invariant states

The construction leads to a physical interpretations of loops of the spin-network graph as a way to capture a distributional magnetic field, analo-gously to what happens in the Aharonov-Bohm effect

The interest in the construction lies in its relation with LQG/Spin Foams andin the perspectives it may open

E. Bianchi (CPT, Marseille) Loop Quantum Gravity a la Aharonov-Bohm ILQGS 21 / 21